HW13
Sujit Thakur
11/23/2021
Q8.2)
We are running \(2^{4-1}\) design where K=4 and P=1. Our defining relationship is I = ABCD and I=-ABCD is this case.
des.res3<-FrF2(nfactors=4,resolution=4,randomize=FALSE)
des.res3## A B C D
## 1 -1 -1 -1 -1
## 2 1 -1 -1 1
## 3 -1 1 -1 1
## 4 1 1 -1 -1
## 5 -1 -1 1 1
## 6 1 -1 1 -1
## 7 -1 1 1 -1
## 8 1 1 1 1
## class=design, type= FrF2response <- c(7.037,16.867,13.876,17.273,11.846,4.368,9.36,15.653)
des.resp1 <- add.response(des.res3,response)
aliasprint(des.resp1)## $legend
## [1] A=A B=B C=C D=D
##
## $main
## character(0)
##
## $fi2
## [1] AB=CD AC=BD AD=BCWe can see that as its 4th resolution , hence the main effects doesnot alias with each other and also not with two factors . But the two factor effects aliased with other two factor.
Following is our design summary
summary(des.resp1)## Call:
## FrF2(nfactors = 4, resolution = 4, randomize = FALSE)
##
## Experimental design of type FrF2
## 8 runs
##
## Factor settings (scale ends):
## A B C D
## 1 -1 -1 -1 -1
## 2 1 1 1 1
##
## Responses:
## [1] response
##
## Design generating information:
## $legend
## [1] A=A B=B C=C D=D
##
## $generators
## [1] D=ABC
##
##
## Alias structure:
## $fi2
## [1] AB=CD AC=BD AD=BC
##
##
## The design itself:
## A B C D response
## 1 -1 -1 -1 -1 7.037
## 2 1 -1 -1 1 16.867
## 3 -1 1 -1 1 13.876
## 4 1 1 -1 -1 17.273
## 5 -1 -1 1 1 11.846
## 6 1 -1 1 -1 4.368
## 7 -1 1 1 -1 9.360
## 8 1 1 1 1 15.653
## class=design, type= FrF2DanielPlot(des.resp1)
We can see from above daniel plot we can see that none of the factors are significant .As all points fall on fairly straigth line
Hence none of the factor effects crack length
Q8.24)
Here we are running \(2^{5-1}\) design where K=5 and P=1. Our defining relation is I = ABCDE and I = - ABCDE is this case.
des.res <- FrF2(nfactors = 5, resolution = 5 ,randomize = FALSE)
aliasprint(des.res)## $legend
## [1] A=A B=B C=C D=D E=E
##
## [[2]]
## [1] no aliasing among main effects and 2fisFollowing is our design summary
summary(des.res)## Call:
## FrF2(nfactors = 5, resolution = 5, randomize = FALSE)
##
## Experimental design of type FrF2
## 16 runs
##
## Factor settings (scale ends):
## A B C D E
## 1 -1 -1 -1 -1 -1
## 2 1 1 1 1 1
##
## Design generating information:
## $legend
## [1] A=A B=B C=C D=D E=E
##
## $generators
## [1] E=ABCD
##
##
## Alias structure:
## [[1]]
## [1] no aliasing among main effects and 2fis
##
##
## The design itself:
## A B C D E
## 1 -1 -1 -1 -1 1
## 2 1 -1 -1 -1 -1
## 3 -1 1 -1 -1 -1
## 4 1 1 -1 -1 1
## 5 -1 -1 1 -1 -1
## 6 1 -1 1 -1 1
## 7 -1 1 1 -1 1
## 8 1 1 1 -1 -1
## 9 -1 -1 -1 1 -1
## 10 1 -1 -1 1 1
## 11 -1 1 -1 1 1
## 12 1 1 -1 1 -1
## 13 -1 -1 1 1 1
## 14 1 -1 1 1 -1
## 15 -1 1 1 1 -1
## 16 1 1 1 1 1
## class=design, type= FrF2AB factor combinations are confounded with blocks.
AB <- c("+","-","-","+","+","-","-","+","+","-","-","+","+","-","-","+")
Block <- c(1,2,2,1,1,2,2,1,1,2,2,1,1,2,2,1)
Data <- data.frame(des.res,AB,Block)
Data## A B C D E AB Block
## 1 -1 -1 -1 -1 1 + 1
## 2 1 -1 -1 -1 -1 - 2
## 3 -1 1 -1 -1 -1 - 2
## 4 1 1 -1 -1 1 + 1
## 5 -1 -1 1 -1 -1 + 1
## 6 1 -1 1 -1 1 - 2
## 7 -1 1 1 -1 1 - 2
## 8 1 1 1 -1 -1 + 1
## 9 -1 -1 -1 1 -1 + 1
## 10 1 -1 -1 1 1 - 2
## 11 -1 1 -1 1 1 - 2
## 12 1 1 -1 1 -1 + 1
## 13 -1 -1 1 1 1 + 1
## 14 1 -1 1 1 -1 - 2
## 15 -1 1 1 1 -1 - 2
## 16 1 1 1 1 1 + 1AB & CDE factor combinations are confounded with blocks.
We can see from above that we have following two factor confounded within two block
Two factor interaction confounded in block 1 =None is present
Two factor interaction confounded in block 2 = None is present
We can see from above that we have following main effects factor confounded within two block
Main effect confounded in Block 2 = A, B
Q8.25)
Here we are running \(2^{7-2}\) design where K7 and P=2. Our defining relation is I =ABCDF I=-ABCDF ; I = ABDEG and I =- ABDEG and we get one generalized interaction as I = CEFG and I =-CEFG in this case.
design <- FrF2(nruns = 32,nfactors=7,blocks = 4,randomize=TRUE)
design## run.no run.no.std.rp Blocks A B C D E F G
## 1 1 22.1.6 1 1 -1 1 -1 1 -1 1
## 2 2 29.1.8 1 1 1 1 -1 -1 1 -1
## 3 3 1.1.1 1 -1 -1 -1 -1 -1 -1 -1
## 4 4 20.1.5 1 1 -1 -1 1 1 1 -1
## 5 5 10.1.3 1 -1 1 -1 -1 1 1 1
## 6 6 7.1.2 1 -1 -1 1 1 -1 1 1
## 7 7 27.1.7 1 1 1 -1 1 -1 -1 1
## 8 8 16.1.4 1 -1 1 1 1 1 -1 -1
## run.no run.no.std.rp Blocks A B C D E F G
## 9 9 9.2.3 2 -1 1 -1 -1 -1 1 1
## 10 10 30.2.8 2 1 1 1 -1 1 1 -1
## 11 11 8.2.2 2 -1 -1 1 1 1 1 1
## 12 12 19.2.5 2 1 -1 -1 1 -1 1 -1
## 13 13 15.2.4 2 -1 1 1 1 -1 -1 -1
## 14 14 28.2.7 2 1 1 -1 1 1 -1 1
## 15 15 21.2.6 2 1 -1 1 -1 -1 -1 1
## 16 16 2.2.1 2 -1 -1 -1 -1 1 -1 -1
## run.no run.no.std.rp Blocks A B C D E F G
## 17 17 24.3.6 3 1 -1 1 1 1 -1 -1
## 18 18 18.3.5 3 1 -1 -1 -1 1 1 1
## 19 19 5.3.2 3 -1 -1 1 -1 -1 1 -1
## 20 20 31.3.8 3 1 1 1 1 -1 1 1
## 21 21 3.3.1 3 -1 -1 -1 1 -1 -1 1
## 22 22 12.3.3 3 -1 1 -1 1 1 1 -1
## 23 23 25.3.7 3 1 1 -1 -1 -1 -1 -1
## 24 24 14.3.4 3 -1 1 1 -1 1 -1 1
## run.no run.no.std.rp Blocks A B C D E F G
## 25 25 32.4.8 4 1 1 1 1 1 1 1
## 26 26 23.4.6 4 1 -1 1 1 -1 -1 -1
## 27 27 26.4.7 4 1 1 -1 -1 1 -1 -1
## 28 28 13.4.4 4 -1 1 1 -1 -1 -1 1
## 29 29 6.4.2 4 -1 -1 1 -1 1 1 -1
## 30 30 11.4.3 4 -1 1 -1 1 -1 1 -1
## 31 31 4.4.1 4 -1 -1 -1 1 1 -1 1
## 32 32 17.4.5 4 1 -1 -1 -1 -1 1 1
## class=design, type= FrF2.blocked
## NOTE: columns run.no and run.no.std.rp are annotation,
## not part of the data framesummary(design)## Call:
## FrF2(nruns = 32, nfactors = 7, blocks = 4, randomize = TRUE)
##
## Experimental design of type FrF2.blocked
## 32 runs
## blocked design with 4 blocks of size 8
##
## Factor settings (scale ends):
## A B C D E F G
## 1 -1 -1 -1 -1 -1 -1 -1
## 2 1 1 1 1 1 1 1
##
## Design generating information:
## $legend
## [1] A=A B=B C=C D=D E=E F=F G=G
##
## $`generators for design itself`
## [1] F=ABC G=ABD
##
## $`block generators`
## [1] ACD ABE
##
##
## Alias structure:
## $fi2
## [1] AB=CF=DG AC=BF AD=BG AF=BC AG=BD CD=FG CG=DF
##
## Aliased with block main effects:
## [1] none
##
## The design itself:
## run.no run.no.std.rp Blocks A B C D E F G
## 1 1 22.1.6 1 1 -1 1 -1 1 -1 1
## 2 2 29.1.8 1 1 1 1 -1 -1 1 -1
## 3 3 1.1.1 1 -1 -1 -1 -1 -1 -1 -1
## 4 4 20.1.5 1 1 -1 -1 1 1 1 -1
## 5 5 10.1.3 1 -1 1 -1 -1 1 1 1
## 6 6 7.1.2 1 -1 -1 1 1 -1 1 1
## 7 7 27.1.7 1 1 1 -1 1 -1 -1 1
## 8 8 16.1.4 1 -1 1 1 1 1 -1 -1
## run.no run.no.std.rp Blocks A B C D E F G
## 9 9 9.2.3 2 -1 1 -1 -1 -1 1 1
## 10 10 30.2.8 2 1 1 1 -1 1 1 -1
## 11 11 8.2.2 2 -1 -1 1 1 1 1 1
## 12 12 19.2.5 2 1 -1 -1 1 -1 1 -1
## 13 13 15.2.4 2 -1 1 1 1 -1 -1 -1
## 14 14 28.2.7 2 1 1 -1 1 1 -1 1
## 15 15 21.2.6 2 1 -1 1 -1 -1 -1 1
## 16 16 2.2.1 2 -1 -1 -1 -1 1 -1 -1
## run.no run.no.std.rp Blocks A B C D E F G
## 17 17 24.3.6 3 1 -1 1 1 1 -1 -1
## 18 18 18.3.5 3 1 -1 -1 -1 1 1 1
## 19 19 5.3.2 3 -1 -1 1 -1 -1 1 -1
## 20 20 31.3.8 3 1 1 1 1 -1 1 1
## 21 21 3.3.1 3 -1 -1 -1 1 -1 -1 1
## 22 22 12.3.3 3 -1 1 -1 1 1 1 -1
## 23 23 25.3.7 3 1 1 -1 -1 -1 -1 -1
## 24 24 14.3.4 3 -1 1 1 -1 1 -1 1
## run.no run.no.std.rp Blocks A B C D E F G
## 25 25 32.4.8 4 1 1 1 1 1 1 1
## 26 26 23.4.6 4 1 -1 1 1 -1 -1 -1
## 27 27 26.4.7 4 1 1 -1 -1 1 -1 -1
## 28 28 13.4.4 4 -1 1 1 -1 -1 -1 1
## 29 29 6.4.2 4 -1 -1 1 -1 1 1 -1
## 30 30 11.4.3 4 -1 1 -1 1 -1 1 -1
## 31 31 4.4.1 4 -1 -1 -1 1 1 -1 1
## 32 32 17.4.5 4 1 -1 -1 -1 -1 1 1
## class=design, type= FrF2.blocked
## NOTE: columns run.no and run.no.std.rp are annotation,
## not part of the data frameBlocks are confounded with ACE , BFG and ABCEFG
No main effects or two factor interactions are confounded with the blocks.
Q8.28)
Ltemp <- c(rep(c("-1","1"),8))
Ltime <- c(rep(c("-1","-1","1","1"),4))
LPres <- c(rep(c("-1","-1","-1","-1","1","1","1","1"),2))
Ftemp <- c(rep(c("-1","-1","-1","-1","-1","-1","-1","-1","1","1","1","1","1","1","1","1"),1))
Fctime <- c(rep(c("-1","1","1","-1","1","-1","-1","1"),2))
Fdpoint <- c(rep(c("-1","1","-1","1","1","-1","1","-1"),2))
resp <- c(0.0167,0.0062,0.0041,0.0073,0.0047,0.0219,0.0121,0.0255,0.0032,0.0078,0.0043,0.0186,0.011,0.0065,0.0155,0.0093,0.0128,0.0066,0.0043,0.0081,0.0047,0.0258,0.009,0.025,0.0023,0.0158,0.0027,0.0137,0.0086,0.0109,0.0158,0.0124,0.0149,0.0044,0.0042,0.0039,0.004,0.0147,0.0092,0.0226,0.0077,0.006,0.0028,0.0158,0.0101,0.0126,0.0145,0.011,0.0185,0.002,0.005,0.003,0.0089,0.0296,0.0086,0.0169,0.0069,0.0045,0.0028,0.0159,0.0158,0.0071,0.0145,0.0133)
total <- c(629,192,176,223,223,920,389,900,201,341,126,640,455,371,603,460)
Mean <- c(157.25,48,44,55.75,55.75,230,97.25,225,50.25,85.25,31.5,160,113.75,92.75,150.75,115)
Std <- c(24.418,20.976,4.083,25.025,22.41,63.639,16.029,39.42,26.725,50.341,7.681,20.083,31.12,29.51,6.75,17.45
)
dat <- cbind(Ltemp,Ltime,LPres,Ftemp,Fctime,Fdpoint,resp,total,Mean,Std)
dat <- as.data.frame(dat)
dat## Ltemp Ltime LPres Ftemp Fctime Fdpoint resp total Mean Std
## 1 -1 -1 -1 -1 -1 -1 0.0167 629 157.25 24.418
## 2 1 -1 -1 -1 1 1 0.0062 192 48 20.976
## 3 -1 1 -1 -1 1 -1 0.0041 176 44 4.083
## 4 1 1 -1 -1 -1 1 0.0073 223 55.75 25.025
## 5 -1 -1 1 -1 1 1 0.0047 223 55.75 22.41
## 6 1 -1 1 -1 -1 -1 0.0219 920 230 63.639
## 7 -1 1 1 -1 -1 1 0.0121 389 97.25 16.029
## 8 1 1 1 -1 1 -1 0.0255 900 225 39.42
## 9 -1 -1 -1 1 -1 -1 0.0032 201 50.25 26.725
## 10 1 -1 -1 1 1 1 0.0078 341 85.25 50.341
## 11 -1 1 -1 1 1 -1 0.0043 126 31.5 7.681
## 12 1 1 -1 1 -1 1 0.0186 640 160 20.083
## 13 -1 -1 1 1 1 1 0.011 455 113.75 31.12
## 14 1 -1 1 1 -1 -1 0.0065 371 92.75 29.51
## 15 -1 1 1 1 -1 1 0.0155 603 150.75 6.75
## 16 1 1 1 1 1 -1 0.0093 460 115 17.45
## 17 -1 -1 -1 -1 -1 -1 0.0128 629 157.25 24.418
## 18 1 -1 -1 -1 1 1 0.0066 192 48 20.976
## 19 -1 1 -1 -1 1 -1 0.0043 176 44 4.083
## 20 1 1 -1 -1 -1 1 0.0081 223 55.75 25.025
## 21 -1 -1 1 -1 1 1 0.0047 223 55.75 22.41
## 22 1 -1 1 -1 -1 -1 0.0258 920 230 63.639
## 23 -1 1 1 -1 -1 1 0.009 389 97.25 16.029
## 24 1 1 1 -1 1 -1 0.025 900 225 39.42
## 25 -1 -1 -1 1 -1 -1 0.0023 201 50.25 26.725
## 26 1 -1 -1 1 1 1 0.0158 341 85.25 50.341
## 27 -1 1 -1 1 1 -1 0.0027 126 31.5 7.681
## 28 1 1 -1 1 -1 1 0.0137 640 160 20.083
## 29 -1 -1 1 1 1 1 0.0086 455 113.75 31.12
## 30 1 -1 1 1 -1 -1 0.0109 371 92.75 29.51
## 31 -1 1 1 1 -1 1 0.0158 603 150.75 6.75
## 32 1 1 1 1 1 -1 0.0124 460 115 17.45
## 33 -1 -1 -1 -1 -1 -1 0.0149 629 157.25 24.418
## 34 1 -1 -1 -1 1 1 0.0044 192 48 20.976
## 35 -1 1 -1 -1 1 -1 0.0042 176 44 4.083
## 36 1 1 -1 -1 -1 1 0.0039 223 55.75 25.025
## 37 -1 -1 1 -1 1 1 0.004 223 55.75 22.41
## 38 1 -1 1 -1 -1 -1 0.0147 920 230 63.639
## 39 -1 1 1 -1 -1 1 0.0092 389 97.25 16.029
## 40 1 1 1 -1 1 -1 0.0226 900 225 39.42
## 41 -1 -1 -1 1 -1 -1 0.0077 201 50.25 26.725
## 42 1 -1 -1 1 1 1 0.006 341 85.25 50.341
## 43 -1 1 -1 1 1 -1 0.0028 126 31.5 7.681
## 44 1 1 -1 1 -1 1 0.0158 640 160 20.083
## 45 -1 -1 1 1 1 1 0.0101 455 113.75 31.12
## 46 1 -1 1 1 -1 -1 0.0126 371 92.75 29.51
## 47 -1 1 1 1 -1 1 0.0145 603 150.75 6.75
## 48 1 1 1 1 1 -1 0.011 460 115 17.45
## 49 -1 -1 -1 -1 -1 -1 0.0185 629 157.25 24.418
## 50 1 -1 -1 -1 1 1 0.002 192 48 20.976
## 51 -1 1 -1 -1 1 -1 0.005 176 44 4.083
## 52 1 1 -1 -1 -1 1 0.003 223 55.75 25.025
## 53 -1 -1 1 -1 1 1 0.0089 223 55.75 22.41
## 54 1 -1 1 -1 -1 -1 0.0296 920 230 63.639
## 55 -1 1 1 -1 -1 1 0.0086 389 97.25 16.029
## 56 1 1 1 -1 1 -1 0.0169 900 225 39.42
## 57 -1 -1 -1 1 -1 -1 0.0069 201 50.25 26.725
## 58 1 -1 -1 1 1 1 0.0045 341 85.25 50.341
## 59 -1 1 -1 1 1 -1 0.0028 126 31.5 7.681
## 60 1 1 -1 1 -1 1 0.0159 640 160 20.083
## 61 -1 -1 1 1 1 1 0.0158 455 113.75 31.12
## 62 1 -1 1 1 -1 -1 0.0071 371 92.75 29.51
## 63 -1 1 1 1 -1 1 0.0145 603 150.75 6.75
## 64 1 1 1 1 1 -1 0.0133 460 115 17.45Answer a) The above design uses \(2^{6-2}\) design with 16 runs at resolution 4
Answer b) following are alias relationship
des.res <- FrF2(nfactors = 6,resolution = 4 , randomize = TRUE)
aliasprint(des.res)## $legend
## [1] A=A B=B C=C D=D E=E F=F
##
## $main
## character(0)
##
## $fi2
## [1] AB=CE=DF AC=BE AD=BF AE=BC AF=BD CD=EF CF=DEFrom table 8.14 from Montgomery we can see design generator is E=(+- ABC) and F=(+-BCD) that is I = ABCE , I = BCDF and generalized interaction is I =ADEF
Answer c)
Ltemp <- c(rep(c(-1,1),8))
Ltime <- c(rep(c(-1,-1,1,1),4))
LPres <- c(rep(c(-1,-1,-1,-1,1,1,1,1),2))
Ftemp <- c(rep(c(-1,-1,-1,-1,-1,-1,-1,-1,1,1,1,1,1,1,1,1),1))
Fctime <- c(rep(c(-1,1,1,-1,1,-1,-1,1),2))
Fdpoint <- c(rep(c(-1,1,-1,1,1,-1,1,-1),2))
resp <- c(0.0167,0.0062,0.0041,0.0073,0.0047,0.0219,0.0121,0.0255,0.0032,0.0078,0.0043,0.0186,0.011,0.0065,0.0155,0.0093,0.0128,0.0066,0.0043,0.0081,0.0047,0.0258,0.009,0.025,0.0023,0.0158,0.0027,0.0137,0.0086,0.0109,0.0158,0.0124,0.0149,0.0044,0.0042,0.0039,0.004,0.0147,0.0092,0.0226,0.0077,0.006,0.0028,0.0158,0.0101,0.0126,0.0145,0.011,0.0185,0.002,0.005,0.003,0.0089,0.0296,0.0086,0.0169,0.0069,0.0045,0.0028,0.0159,0.0158,0.0071,0.0145,0.0133)
total <- c(629,192,176,223,223,920,389,900,201,341,126,640,455,371,603,460)
Mean <- c(157.25,48,44,55.75,55.75,230,97.25,225,50.25,85.25,31.5,160,113.75,92.75,150.75,115)
Std <- c(24.418,20.976,4.083,25.025,22.41,63.639,16.029,39.42,26.725,50.341,7.681,20.083,31.12,29.51,6.75,17.45
)
dat <- cbind(Ltemp,Ltime,LPres,Ftemp,Fctime,Fdpoint,resp,total,Mean,Std)
dat <- as.data.frame(dat)
dat## Ltemp Ltime LPres Ftemp Fctime Fdpoint resp total Mean Std
## 1 -1 -1 -1 -1 -1 -1 0.0167 629 157.25 24.418
## 2 1 -1 -1 -1 1 1 0.0062 192 48.00 20.976
## 3 -1 1 -1 -1 1 -1 0.0041 176 44.00 4.083
## 4 1 1 -1 -1 -1 1 0.0073 223 55.75 25.025
## 5 -1 -1 1 -1 1 1 0.0047 223 55.75 22.410
## 6 1 -1 1 -1 -1 -1 0.0219 920 230.00 63.639
## 7 -1 1 1 -1 -1 1 0.0121 389 97.25 16.029
## 8 1 1 1 -1 1 -1 0.0255 900 225.00 39.420
## 9 -1 -1 -1 1 -1 -1 0.0032 201 50.25 26.725
## 10 1 -1 -1 1 1 1 0.0078 341 85.25 50.341
## 11 -1 1 -1 1 1 -1 0.0043 126 31.50 7.681
## 12 1 1 -1 1 -1 1 0.0186 640 160.00 20.083
## 13 -1 -1 1 1 1 1 0.0110 455 113.75 31.120
## 14 1 -1 1 1 -1 -1 0.0065 371 92.75 29.510
## 15 -1 1 1 1 -1 1 0.0155 603 150.75 6.750
## 16 1 1 1 1 1 -1 0.0093 460 115.00 17.450
## 17 -1 -1 -1 -1 -1 -1 0.0128 629 157.25 24.418
## 18 1 -1 -1 -1 1 1 0.0066 192 48.00 20.976
## 19 -1 1 -1 -1 1 -1 0.0043 176 44.00 4.083
## 20 1 1 -1 -1 -1 1 0.0081 223 55.75 25.025
## 21 -1 -1 1 -1 1 1 0.0047 223 55.75 22.410
## 22 1 -1 1 -1 -1 -1 0.0258 920 230.00 63.639
## 23 -1 1 1 -1 -1 1 0.0090 389 97.25 16.029
## 24 1 1 1 -1 1 -1 0.0250 900 225.00 39.420
## 25 -1 -1 -1 1 -1 -1 0.0023 201 50.25 26.725
## 26 1 -1 -1 1 1 1 0.0158 341 85.25 50.341
## 27 -1 1 -1 1 1 -1 0.0027 126 31.50 7.681
## 28 1 1 -1 1 -1 1 0.0137 640 160.00 20.083
## 29 -1 -1 1 1 1 1 0.0086 455 113.75 31.120
## 30 1 -1 1 1 -1 -1 0.0109 371 92.75 29.510
## 31 -1 1 1 1 -1 1 0.0158 603 150.75 6.750
## 32 1 1 1 1 1 -1 0.0124 460 115.00 17.450
## 33 -1 -1 -1 -1 -1 -1 0.0149 629 157.25 24.418
## 34 1 -1 -1 -1 1 1 0.0044 192 48.00 20.976
## 35 -1 1 -1 -1 1 -1 0.0042 176 44.00 4.083
## 36 1 1 -1 -1 -1 1 0.0039 223 55.75 25.025
## 37 -1 -1 1 -1 1 1 0.0040 223 55.75 22.410
## 38 1 -1 1 -1 -1 -1 0.0147 920 230.00 63.639
## 39 -1 1 1 -1 -1 1 0.0092 389 97.25 16.029
## 40 1 1 1 -1 1 -1 0.0226 900 225.00 39.420
## 41 -1 -1 -1 1 -1 -1 0.0077 201 50.25 26.725
## 42 1 -1 -1 1 1 1 0.0060 341 85.25 50.341
## 43 -1 1 -1 1 1 -1 0.0028 126 31.50 7.681
## 44 1 1 -1 1 -1 1 0.0158 640 160.00 20.083
## 45 -1 -1 1 1 1 1 0.0101 455 113.75 31.120
## 46 1 -1 1 1 -1 -1 0.0126 371 92.75 29.510
## 47 -1 1 1 1 -1 1 0.0145 603 150.75 6.750
## 48 1 1 1 1 1 -1 0.0110 460 115.00 17.450
## 49 -1 -1 -1 -1 -1 -1 0.0185 629 157.25 24.418
## 50 1 -1 -1 -1 1 1 0.0020 192 48.00 20.976
## 51 -1 1 -1 -1 1 -1 0.0050 176 44.00 4.083
## 52 1 1 -1 -1 -1 1 0.0030 223 55.75 25.025
## 53 -1 -1 1 -1 1 1 0.0089 223 55.75 22.410
## 54 1 -1 1 -1 -1 -1 0.0296 920 230.00 63.639
## 55 -1 1 1 -1 -1 1 0.0086 389 97.25 16.029
## 56 1 1 1 -1 1 -1 0.0169 900 225.00 39.420
## 57 -1 -1 -1 1 -1 -1 0.0069 201 50.25 26.725
## 58 1 -1 -1 1 1 1 0.0045 341 85.25 50.341
## 59 -1 1 -1 1 1 -1 0.0028 126 31.50 7.681
## 60 1 1 -1 1 -1 1 0.0159 640 160.00 20.083
## 61 -1 -1 1 1 1 1 0.0158 455 113.75 31.120
## 62 1 -1 1 1 -1 -1 0.0071 371 92.75 29.510
## 63 -1 1 1 1 -1 1 0.0145 603 150.75 6.750
## 64 1 1 1 1 1 -1 0.0133 460 115.00 17.450dat$Ltemp <- as.fixed(dat$Ltemp)
dat$Ltime <- as.fixed(dat$Ltime)
dat$LPres <- as.fixed(dat$LPres)
dat$Ftemp <- as.fixed(dat$Ftemp)
dat$Fdpoint <- as.fixed(dat$Fdpoint)
dat$Fctime <- as.fixed(dat$Fctime)
model <- aov(resp~Ltemp*Ltime*LPres*Ftemp*Fctime*Fdpoint,data = dat)
summary(model)## Df Sum Sq Mean Sq F value Pr(>F)
## Ltemp 1 0.0002422 0.0002422 27.793 3.17e-06 ***
## Ltime 1 0.0000053 0.0000053 0.614 0.43725
## LPres 1 0.0005023 0.0005023 57.644 9.14e-10 ***
## Ftemp 1 0.0000323 0.0000323 3.712 0.05995 .
## Fctime 1 0.0001901 0.0001901 21.815 2.45e-05 ***
## Fdpoint 1 0.0000803 0.0000803 9.218 0.00387 **
## Ltemp:Ltime 1 0.0000587 0.0000587 6.738 0.01249 *
## Ltime:LPres 1 0.0000527 0.0000527 6.053 0.01754 *
## Ltemp:Ftemp 1 0.0000239 0.0000239 2.741 0.10431
## Ltime:Ftemp 1 0.0000849 0.0000849 9.739 0.00305 **
## LPres:Ftemp 1 0.0000622 0.0000622 7.139 0.01027 *
## Ftemp:Fctime 1 0.0000088 0.0000088 1.007 0.32062
## Ftemp:Fdpoint 1 0.0009602 0.0009602 110.192 5.05e-14 ***
## Ltemp:Ltime:Ftemp 1 0.0000000 0.0000000 0.005 0.94291
## Ltime:LPres:Ftemp 1 0.0000481 0.0000481 5.523 0.02293 *
## Residuals 48 0.0004183 0.0000087
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1Above analysis shows some main effects and higher interactions that are significant and that would affect average camber.
Process Variable which will affect average camber is A,C,E and F
Answer d)
stdcamber <- c(24.418,20.976,4.083,25.025,22.41,63.639,16.029,39.42,26.725,50.341,7.681,20.083,31.12,29.51,6.75,17.45)
var <- stdcamber^2
A <- c(rep(c("-1","1"),8))
B <- c(rep(c("-1","-1","1","1"),4))
C <- c(rep(c("-1","-1","-1","-1","1","1","1","1"),2))
D <- c(rep(c("-1","-1","-1","-1","-1","-1","-1","-1","1","1","1","1","1","1","1","1"),1))
E <- c(rep(c("-1","1","1","-1","1","-1","-1","1"),2))
F<- c(rep(c("-1","1","-1","1","1","-1","1","-1"),2))
data <- data.frame(A,B,C,D,E,F,var)
model <- lm(stdcamber~A*B*C*D*E*F,data = data)
DanielPlot(model)
## from above daniel plot it is clear that it is Ltemp (A) and Ltime(B) which is significantly affecting standard deviation
model <- aov(stdcamber~A+B,data = data)
summary(model)## Df Sum Sq Mean Sq F value Pr(>F)
## A 1 1012 1012 8.505 0.01202 *
## B 1 1099 1099 9.241 0.00948 **
## Residuals 13 1546 119
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1from above Anova test now its clear that yes Ltemp and Ltime has p value < 0.05 , which makes our claim correct that only Ltemp and Ltime are affecting standard deviation of camber
Answer e)
Ltemp <- c(rep(c("-1","1"),8))
Ltime <- c(rep(c("-1","-1","1","1"),4))
LPres <- c(rep(c("-1","-1","-1","-1","1","1","1","1"),2))
Ftemp <- c(rep(c("-1","-1","-1","-1","-1","-1","-1","-1","1","1","1","1","1","1","1","1"),1))
Fctime <- c(rep(c("-1","1","1","-1","1","-1","-1","1"),2))
Fdpoint <- c(rep(c("-1","1","-1","1","1","-1","1","-1"),2))
resp <- c(0.0167,0.0062,0.0041,0.0073,0.0047,0.0219,0.0121,0.0255,0.0032,0.0078,0.0043,0.0186,0.011,0.0065,0.0155,0.0093,0.0128,0.0066,0.0043,0.0081,0.0047,0.0258,0.009,0.025,0.0023,0.0158,0.0027,0.0137,0.0086,0.0109,0.0158,0.0124,0.0149,0.0044,0.0042,0.0039,0.004,0.0147,0.0092,0.0226,0.0077,0.006,0.0028,0.0158,0.0101,0.0126,0.0145,0.011,0.0185,0.002,0.005,0.003,0.0089,0.0296,0.0086,0.0169,0.0069,0.0045,0.0028,0.0159,0.0158,0.0071,0.0145,0.0133)
total <- c(629,192,176,223,223,920,389,900,201,341,126,640,455,371,603,460)
Mean <- c(157.25,48,44,55.75,55.75,230,97.25,225,50.25,85.25,31.5,160,113.75,92.75,150.75,115)
Std <- c(24.418,20.976,4.083,25.025,22.41,63.639,16.029,39.42,26.725,50.341,7.681,20.083,31.12,29.51,6.75,17.45
)
dat <- cbind(Ltemp,Ltime,LPres,Ftemp,Fctime,Fdpoint,resp,total,Mean,Std)
dat <- as.data.frame(dat)
model <- lm(dat$resp~dat$Ltemp*dat$Ltime*dat$LPres*dat$Ftemp*dat$Fctime*dat$Fdpoint)
coef(model)## (Intercept)
## 0.01572500
## dat$Ltemp1
## 0.00325000
## dat$Ltime1
## -0.00713750
## dat$LPres1
## 0.00402500
## dat$Ftemp1
## -0.01070000
## dat$Fctime1
## -0.00418750
## dat$Fdpoint1
## -0.00998750
## dat$Ltemp1:dat$Ltime1
## 0.00372500
## dat$Ltemp1:dat$LPres1
## NA
## dat$Ltime1:dat$LPres1
## 0.00710000
## dat$Ltemp1:dat$Ftemp1
## -0.00255000
## dat$Ltime1:dat$Ftemp1
## 0.00796875
## dat$LPres1:dat$Ftemp1
## -0.00047500
## dat$Ltemp1:dat$Fctime1
## NA
## dat$Ltime1:dat$Fctime1
## NA
## dat$LPres1:dat$Fctime1
## NA
## dat$Ftemp1:dat$Fctime1
## 0.00148125
## dat$Ltemp1:dat$Fdpoint1
## NA
## dat$Ltime1:dat$Fdpoint1
## NA
## dat$LPres1:dat$Fdpoint1
## NA
## dat$Ftemp1:dat$Fdpoint1
## 0.01549375
## dat$Fctime1:dat$Fdpoint1
## NA
## dat$Ltemp1:dat$Ltime1:dat$LPres1
## NA
## dat$Ltemp1:dat$Ltime1:dat$Ftemp1
## 0.00021250
## dat$Ltemp1:dat$LPres1:dat$Ftemp1
## NA
## dat$Ltime1:dat$LPres1:dat$Ftemp1
## -0.00693750
## dat$Ltemp1:dat$Ltime1:dat$Fctime1
## NA
## dat$Ltemp1:dat$LPres1:dat$Fctime1
## NA
## dat$Ltime1:dat$LPres1:dat$Fctime1
## NA
## dat$Ltemp1:dat$Ftemp1:dat$Fctime1
## NA
## dat$Ltime1:dat$Ftemp1:dat$Fctime1
## NA
## dat$LPres1:dat$Ftemp1:dat$Fctime1
## NA
## dat$Ltemp1:dat$Ltime1:dat$Fdpoint1
## NA
## dat$Ltemp1:dat$LPres1:dat$Fdpoint1
## NA
## dat$Ltime1:dat$LPres1:dat$Fdpoint1
## NA
## dat$Ltemp1:dat$Ftemp1:dat$Fdpoint1
## NA
## dat$Ltime1:dat$Ftemp1:dat$Fdpoint1
## NA
## dat$LPres1:dat$Ftemp1:dat$Fdpoint1
## NA
## dat$Ltemp1:dat$Fctime1:dat$Fdpoint1
## NA
## dat$Ltime1:dat$Fctime1:dat$Fdpoint1
## NA
## dat$LPres1:dat$Fctime1:dat$Fdpoint1
## NA
## dat$Ftemp1:dat$Fctime1:dat$Fdpoint1
## NA
## dat$Ltemp1:dat$Ltime1:dat$LPres1:dat$Ftemp1
## NA
## dat$Ltemp1:dat$Ltime1:dat$LPres1:dat$Fctime1
## NA
## dat$Ltemp1:dat$Ltime1:dat$Ftemp1:dat$Fctime1
## NA
## dat$Ltemp1:dat$LPres1:dat$Ftemp1:dat$Fctime1
## NA
## dat$Ltime1:dat$LPres1:dat$Ftemp1:dat$Fctime1
## NA
## dat$Ltemp1:dat$Ltime1:dat$LPres1:dat$Fdpoint1
## NA
## dat$Ltemp1:dat$Ltime1:dat$Ftemp1:dat$Fdpoint1
## NA
## dat$Ltemp1:dat$LPres1:dat$Ftemp1:dat$Fdpoint1
## NA
## dat$Ltime1:dat$LPres1:dat$Ftemp1:dat$Fdpoint1
## NA
## dat$Ltemp1:dat$Ltime1:dat$Fctime1:dat$Fdpoint1
## NA
## dat$Ltemp1:dat$LPres1:dat$Fctime1:dat$Fdpoint1
## NA
## dat$Ltime1:dat$LPres1:dat$Fctime1:dat$Fdpoint1
## NA
## dat$Ltemp1:dat$Ftemp1:dat$Fctime1:dat$Fdpoint1
## NA
## dat$Ltime1:dat$Ftemp1:dat$Fctime1:dat$Fdpoint1
## NA
## dat$LPres1:dat$Ftemp1:dat$Fctime1:dat$Fdpoint1
## NA
## dat$Ltemp1:dat$Ltime1:dat$LPres1:dat$Ftemp1:dat$Fctime1
## NA
## dat$Ltemp1:dat$Ltime1:dat$LPres1:dat$Ftemp1:dat$Fdpoint1
## NA
## dat$Ltemp1:dat$Ltime1:dat$LPres1:dat$Fctime1:dat$Fdpoint1
## NA
## dat$Ltemp1:dat$Ltime1:dat$Ftemp1:dat$Fctime1:dat$Fdpoint1
## NA
## dat$Ltemp1:dat$LPres1:dat$Ftemp1:dat$Fctime1:dat$Fdpoint1
## NA
## dat$Ltime1:dat$LPres1:dat$Ftemp1:dat$Fctime1:dat$Fdpoint1
## NA
## dat$Ltemp1:dat$Ltime1:dat$LPres1:dat$Ftemp1:dat$Fctime1:dat$Fdpoint1
## NAsummary(model)##
## Call:
## lm.default(formula = dat$resp ~ dat$Ltemp * dat$Ltime * dat$LPres *
## dat$Ftemp * dat$Fctime * dat$Fdpoint)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.008300 -0.001350 -0.000350 0.001744 0.007275
##
## Coefficients: (48 not defined because of singularities)
## Estimate
## (Intercept) 0.0157250
## dat$Ltemp1 0.0032500
## dat$Ltime1 -0.0071375
## dat$LPres1 0.0040250
## dat$Ftemp1 -0.0107000
## dat$Fctime1 -0.0041875
## dat$Fdpoint1 -0.0099875
## dat$Ltemp1:dat$Ltime1 0.0037250
## dat$Ltemp1:dat$LPres1 NA
## dat$Ltime1:dat$LPres1 0.0071000
## dat$Ltemp1:dat$Ftemp1 -0.0025500
## dat$Ltime1:dat$Ftemp1 0.0079688
## dat$LPres1:dat$Ftemp1 -0.0004750
## dat$Ltemp1:dat$Fctime1 NA
## dat$Ltime1:dat$Fctime1 NA
## dat$LPres1:dat$Fctime1 NA
## dat$Ftemp1:dat$Fctime1 0.0014813
## dat$Ltemp1:dat$Fdpoint1 NA
## dat$Ltime1:dat$Fdpoint1 NA
## dat$LPres1:dat$Fdpoint1 NA
## dat$Ftemp1:dat$Fdpoint1 0.0154937
## dat$Fctime1:dat$Fdpoint1 NA
## dat$Ltemp1:dat$Ltime1:dat$LPres1 NA
## dat$Ltemp1:dat$Ltime1:dat$Ftemp1 0.0002125
## dat$Ltemp1:dat$LPres1:dat$Ftemp1 NA
## dat$Ltime1:dat$LPres1:dat$Ftemp1 -0.0069375
## dat$Ltemp1:dat$Ltime1:dat$Fctime1 NA
## dat$Ltemp1:dat$LPres1:dat$Fctime1 NA
## dat$Ltime1:dat$LPres1:dat$Fctime1 NA
## dat$Ltemp1:dat$Ftemp1:dat$Fctime1 NA
## dat$Ltime1:dat$Ftemp1:dat$Fctime1 NA
## dat$LPres1:dat$Ftemp1:dat$Fctime1 NA
## dat$Ltemp1:dat$Ltime1:dat$Fdpoint1 NA
## dat$Ltemp1:dat$LPres1:dat$Fdpoint1 NA
## dat$Ltime1:dat$LPres1:dat$Fdpoint1 NA
## dat$Ltemp1:dat$Ftemp1:dat$Fdpoint1 NA
## dat$Ltime1:dat$Ftemp1:dat$Fdpoint1 NA
## dat$LPres1:dat$Ftemp1:dat$Fdpoint1 NA
## dat$Ltemp1:dat$Fctime1:dat$Fdpoint1 NA
## dat$Ltime1:dat$Fctime1:dat$Fdpoint1 NA
## dat$LPres1:dat$Fctime1:dat$Fdpoint1 NA
## dat$Ftemp1:dat$Fctime1:dat$Fdpoint1 NA
## dat$Ltemp1:dat$Ltime1:dat$LPres1:dat$Ftemp1 NA
## dat$Ltemp1:dat$Ltime1:dat$LPres1:dat$Fctime1 NA
## dat$Ltemp1:dat$Ltime1:dat$Ftemp1:dat$Fctime1 NA
## dat$Ltemp1:dat$LPres1:dat$Ftemp1:dat$Fctime1 NA
## dat$Ltime1:dat$LPres1:dat$Ftemp1:dat$Fctime1 NA
## dat$Ltemp1:dat$Ltime1:dat$LPres1:dat$Fdpoint1 NA
## dat$Ltemp1:dat$Ltime1:dat$Ftemp1:dat$Fdpoint1 NA
## dat$Ltemp1:dat$LPres1:dat$Ftemp1:dat$Fdpoint1 NA
## dat$Ltime1:dat$LPres1:dat$Ftemp1:dat$Fdpoint1 NA
## dat$Ltemp1:dat$Ltime1:dat$Fctime1:dat$Fdpoint1 NA
## dat$Ltemp1:dat$LPres1:dat$Fctime1:dat$Fdpoint1 NA
## dat$Ltime1:dat$LPres1:dat$Fctime1:dat$Fdpoint1 NA
## dat$Ltemp1:dat$Ftemp1:dat$Fctime1:dat$Fdpoint1 NA
## dat$Ltime1:dat$Ftemp1:dat$Fctime1:dat$Fdpoint1 NA
## dat$LPres1:dat$Ftemp1:dat$Fctime1:dat$Fdpoint1 NA
## dat$Ltemp1:dat$Ltime1:dat$LPres1:dat$Ftemp1:dat$Fctime1 NA
## dat$Ltemp1:dat$Ltime1:dat$LPres1:dat$Ftemp1:dat$Fdpoint1 NA
## dat$Ltemp1:dat$Ltime1:dat$LPres1:dat$Fctime1:dat$Fdpoint1 NA
## dat$Ltemp1:dat$Ltime1:dat$Ftemp1:dat$Fctime1:dat$Fdpoint1 NA
## dat$Ltemp1:dat$LPres1:dat$Ftemp1:dat$Fctime1:dat$Fdpoint1 NA
## dat$Ltime1:dat$LPres1:dat$Ftemp1:dat$Fctime1:dat$Fdpoint1 NA
## dat$Ltemp1:dat$Ltime1:dat$LPres1:dat$Ftemp1:dat$Fctime1:dat$Fdpoint1 NA
## Std. Error
## (Intercept) 0.0014760
## dat$Ltemp1 0.0014760
## dat$Ltime1 0.0018077
## dat$LPres1 0.0014760
## dat$Ftemp1 0.0020874
## dat$Fctime1 0.0010437
## dat$Fdpoint1 0.0010437
## dat$Ltemp1:dat$Ltime1 0.0020874
## dat$Ltemp1:dat$LPres1 NA
## dat$Ltime1:dat$LPres1 0.0020874
## dat$Ltemp1:dat$Ftemp1 0.0020874
## dat$Ltime1:dat$Ftemp1 0.0025565
## dat$LPres1:dat$Ftemp1 0.0020874
## dat$Ltemp1:dat$Fctime1 NA
## dat$Ltime1:dat$Fctime1 NA
## dat$LPres1:dat$Fctime1 NA
## dat$Ftemp1:dat$Fctime1 0.0014760
## dat$Ltemp1:dat$Fdpoint1 NA
## dat$Ltime1:dat$Fdpoint1 NA
## dat$LPres1:dat$Fdpoint1 NA
## dat$Ftemp1:dat$Fdpoint1 0.0014760
## dat$Fctime1:dat$Fdpoint1 NA
## dat$Ltemp1:dat$Ltime1:dat$LPres1 NA
## dat$Ltemp1:dat$Ltime1:dat$Ftemp1 0.0029520
## dat$Ltemp1:dat$LPres1:dat$Ftemp1 NA
## dat$Ltime1:dat$LPres1:dat$Ftemp1 0.0029520
## dat$Ltemp1:dat$Ltime1:dat$Fctime1 NA
## dat$Ltemp1:dat$LPres1:dat$Fctime1 NA
## dat$Ltime1:dat$LPres1:dat$Fctime1 NA
## dat$Ltemp1:dat$Ftemp1:dat$Fctime1 NA
## dat$Ltime1:dat$Ftemp1:dat$Fctime1 NA
## dat$LPres1:dat$Ftemp1:dat$Fctime1 NA
## dat$Ltemp1:dat$Ltime1:dat$Fdpoint1 NA
## dat$Ltemp1:dat$LPres1:dat$Fdpoint1 NA
## dat$Ltime1:dat$LPres1:dat$Fdpoint1 NA
## dat$Ltemp1:dat$Ftemp1:dat$Fdpoint1 NA
## dat$Ltime1:dat$Ftemp1:dat$Fdpoint1 NA
## dat$LPres1:dat$Ftemp1:dat$Fdpoint1 NA
## dat$Ltemp1:dat$Fctime1:dat$Fdpoint1 NA
## dat$Ltime1:dat$Fctime1:dat$Fdpoint1 NA
## dat$LPres1:dat$Fctime1:dat$Fdpoint1 NA
## dat$Ftemp1:dat$Fctime1:dat$Fdpoint1 NA
## dat$Ltemp1:dat$Ltime1:dat$LPres1:dat$Ftemp1 NA
## dat$Ltemp1:dat$Ltime1:dat$LPres1:dat$Fctime1 NA
## dat$Ltemp1:dat$Ltime1:dat$Ftemp1:dat$Fctime1 NA
## dat$Ltemp1:dat$LPres1:dat$Ftemp1:dat$Fctime1 NA
## dat$Ltime1:dat$LPres1:dat$Ftemp1:dat$Fctime1 NA
## dat$Ltemp1:dat$Ltime1:dat$LPres1:dat$Fdpoint1 NA
## dat$Ltemp1:dat$Ltime1:dat$Ftemp1:dat$Fdpoint1 NA
## dat$Ltemp1:dat$LPres1:dat$Ftemp1:dat$Fdpoint1 NA
## dat$Ltime1:dat$LPres1:dat$Ftemp1:dat$Fdpoint1 NA
## dat$Ltemp1:dat$Ltime1:dat$Fctime1:dat$Fdpoint1 NA
## dat$Ltemp1:dat$LPres1:dat$Fctime1:dat$Fdpoint1 NA
## dat$Ltime1:dat$LPres1:dat$Fctime1:dat$Fdpoint1 NA
## dat$Ltemp1:dat$Ftemp1:dat$Fctime1:dat$Fdpoint1 NA
## dat$Ltime1:dat$Ftemp1:dat$Fctime1:dat$Fdpoint1 NA
## dat$LPres1:dat$Ftemp1:dat$Fctime1:dat$Fdpoint1 NA
## dat$Ltemp1:dat$Ltime1:dat$LPres1:dat$Ftemp1:dat$Fctime1 NA
## dat$Ltemp1:dat$Ltime1:dat$LPres1:dat$Ftemp1:dat$Fdpoint1 NA
## dat$Ltemp1:dat$Ltime1:dat$LPres1:dat$Fctime1:dat$Fdpoint1 NA
## dat$Ltemp1:dat$Ltime1:dat$Ftemp1:dat$Fctime1:dat$Fdpoint1 NA
## dat$Ltemp1:dat$LPres1:dat$Ftemp1:dat$Fctime1:dat$Fdpoint1 NA
## dat$Ltime1:dat$LPres1:dat$Ftemp1:dat$Fctime1:dat$Fdpoint1 NA
## dat$Ltemp1:dat$Ltime1:dat$LPres1:dat$Ftemp1:dat$Fctime1:dat$Fdpoint1 NA
## t value
## (Intercept) 10.654
## dat$Ltemp1 2.202
## dat$Ltime1 -3.948
## dat$LPres1 2.727
## dat$Ftemp1 -5.126
## dat$Fctime1 -4.012
## dat$Fdpoint1 -9.570
## dat$Ltemp1:dat$Ltime1 1.785
## dat$Ltemp1:dat$LPres1 NA
## dat$Ltime1:dat$LPres1 3.401
## dat$Ltemp1:dat$Ftemp1 -1.222
## dat$Ltime1:dat$Ftemp1 3.117
## dat$LPres1:dat$Ftemp1 -0.228
## dat$Ltemp1:dat$Fctime1 NA
## dat$Ltime1:dat$Fctime1 NA
## dat$LPres1:dat$Fctime1 NA
## dat$Ftemp1:dat$Fctime1 1.004
## dat$Ltemp1:dat$Fdpoint1 NA
## dat$Ltime1:dat$Fdpoint1 NA
## dat$LPres1:dat$Fdpoint1 NA
## dat$Ftemp1:dat$Fdpoint1 10.497
## dat$Fctime1:dat$Fdpoint1 NA
## dat$Ltemp1:dat$Ltime1:dat$LPres1 NA
## dat$Ltemp1:dat$Ltime1:dat$Ftemp1 0.072
## dat$Ltemp1:dat$LPres1:dat$Ftemp1 NA
## dat$Ltime1:dat$LPres1:dat$Ftemp1 -2.350
## dat$Ltemp1:dat$Ltime1:dat$Fctime1 NA
## dat$Ltemp1:dat$LPres1:dat$Fctime1 NA
## dat$Ltime1:dat$LPres1:dat$Fctime1 NA
## dat$Ltemp1:dat$Ftemp1:dat$Fctime1 NA
## dat$Ltime1:dat$Ftemp1:dat$Fctime1 NA
## dat$LPres1:dat$Ftemp1:dat$Fctime1 NA
## dat$Ltemp1:dat$Ltime1:dat$Fdpoint1 NA
## dat$Ltemp1:dat$LPres1:dat$Fdpoint1 NA
## dat$Ltime1:dat$LPres1:dat$Fdpoint1 NA
## dat$Ltemp1:dat$Ftemp1:dat$Fdpoint1 NA
## dat$Ltime1:dat$Ftemp1:dat$Fdpoint1 NA
## dat$LPres1:dat$Ftemp1:dat$Fdpoint1 NA
## dat$Ltemp1:dat$Fctime1:dat$Fdpoint1 NA
## dat$Ltime1:dat$Fctime1:dat$Fdpoint1 NA
## dat$LPres1:dat$Fctime1:dat$Fdpoint1 NA
## dat$Ftemp1:dat$Fctime1:dat$Fdpoint1 NA
## dat$Ltemp1:dat$Ltime1:dat$LPres1:dat$Ftemp1 NA
## dat$Ltemp1:dat$Ltime1:dat$LPres1:dat$Fctime1 NA
## dat$Ltemp1:dat$Ltime1:dat$Ftemp1:dat$Fctime1 NA
## dat$Ltemp1:dat$LPres1:dat$Ftemp1:dat$Fctime1 NA
## dat$Ltime1:dat$LPres1:dat$Ftemp1:dat$Fctime1 NA
## dat$Ltemp1:dat$Ltime1:dat$LPres1:dat$Fdpoint1 NA
## dat$Ltemp1:dat$Ltime1:dat$Ftemp1:dat$Fdpoint1 NA
## dat$Ltemp1:dat$LPres1:dat$Ftemp1:dat$Fdpoint1 NA
## dat$Ltime1:dat$LPres1:dat$Ftemp1:dat$Fdpoint1 NA
## dat$Ltemp1:dat$Ltime1:dat$Fctime1:dat$Fdpoint1 NA
## dat$Ltemp1:dat$LPres1:dat$Fctime1:dat$Fdpoint1 NA
## dat$Ltime1:dat$LPres1:dat$Fctime1:dat$Fdpoint1 NA
## dat$Ltemp1:dat$Ftemp1:dat$Fctime1:dat$Fdpoint1 NA
## dat$Ltime1:dat$Ftemp1:dat$Fctime1:dat$Fdpoint1 NA
## dat$LPres1:dat$Ftemp1:dat$Fctime1:dat$Fdpoint1 NA
## dat$Ltemp1:dat$Ltime1:dat$LPres1:dat$Ftemp1:dat$Fctime1 NA
## dat$Ltemp1:dat$Ltime1:dat$LPres1:dat$Ftemp1:dat$Fdpoint1 NA
## dat$Ltemp1:dat$Ltime1:dat$LPres1:dat$Fctime1:dat$Fdpoint1 NA
## dat$Ltemp1:dat$Ltime1:dat$Ftemp1:dat$Fctime1:dat$Fdpoint1 NA
## dat$Ltemp1:dat$LPres1:dat$Ftemp1:dat$Fctime1:dat$Fdpoint1 NA
## dat$Ltime1:dat$LPres1:dat$Ftemp1:dat$Fctime1:dat$Fdpoint1 NA
## dat$Ltemp1:dat$Ltime1:dat$LPres1:dat$Ftemp1:dat$Fctime1:dat$Fdpoint1 NA
## Pr(>|t|)
## (Intercept) 3.06e-14
## dat$Ltemp1 0.032509
## dat$Ltime1 0.000257
## dat$LPres1 0.008899
## dat$Ftemp1 5.24e-06
## dat$Fctime1 0.000210
## dat$Fdpoint1 1.05e-12
## dat$Ltemp1:dat$Ltime1 0.080655
## dat$Ltemp1:dat$LPres1 NA
## dat$Ltime1:dat$LPres1 0.001359
## dat$Ltemp1:dat$Ftemp1 0.227809
## dat$Ltime1:dat$Ftemp1 0.003083
## dat$LPres1:dat$Ftemp1 0.820954
## dat$Ltemp1:dat$Fctime1 NA
## dat$Ltime1:dat$Fctime1 NA
## dat$LPres1:dat$Fctime1 NA
## dat$Ftemp1:dat$Fctime1 0.320619
## dat$Ltemp1:dat$Fdpoint1 NA
## dat$Ltime1:dat$Fdpoint1 NA
## dat$LPres1:dat$Fdpoint1 NA
## dat$Ftemp1:dat$Fdpoint1 5.05e-14
## dat$Fctime1:dat$Fdpoint1 NA
## dat$Ltemp1:dat$Ltime1:dat$LPres1 NA
## dat$Ltemp1:dat$Ltime1:dat$Ftemp1 0.942912
## dat$Ltemp1:dat$LPres1:dat$Ftemp1 NA
## dat$Ltime1:dat$LPres1:dat$Ftemp1 0.022926
## dat$Ltemp1:dat$Ltime1:dat$Fctime1 NA
## dat$Ltemp1:dat$LPres1:dat$Fctime1 NA
## dat$Ltime1:dat$LPres1:dat$Fctime1 NA
## dat$Ltemp1:dat$Ftemp1:dat$Fctime1 NA
## dat$Ltime1:dat$Ftemp1:dat$Fctime1 NA
## dat$LPres1:dat$Ftemp1:dat$Fctime1 NA
## dat$Ltemp1:dat$Ltime1:dat$Fdpoint1 NA
## dat$Ltemp1:dat$LPres1:dat$Fdpoint1 NA
## dat$Ltime1:dat$LPres1:dat$Fdpoint1 NA
## dat$Ltemp1:dat$Ftemp1:dat$Fdpoint1 NA
## dat$Ltime1:dat$Ftemp1:dat$Fdpoint1 NA
## dat$LPres1:dat$Ftemp1:dat$Fdpoint1 NA
## dat$Ltemp1:dat$Fctime1:dat$Fdpoint1 NA
## dat$Ltime1:dat$Fctime1:dat$Fdpoint1 NA
## dat$LPres1:dat$Fctime1:dat$Fdpoint1 NA
## dat$Ftemp1:dat$Fctime1:dat$Fdpoint1 NA
## dat$Ltemp1:dat$Ltime1:dat$LPres1:dat$Ftemp1 NA
## dat$Ltemp1:dat$Ltime1:dat$LPres1:dat$Fctime1 NA
## dat$Ltemp1:dat$Ltime1:dat$Ftemp1:dat$Fctime1 NA
## dat$Ltemp1:dat$LPres1:dat$Ftemp1:dat$Fctime1 NA
## dat$Ltime1:dat$LPres1:dat$Ftemp1:dat$Fctime1 NA
## dat$Ltemp1:dat$Ltime1:dat$LPres1:dat$Fdpoint1 NA
## dat$Ltemp1:dat$Ltime1:dat$Ftemp1:dat$Fdpoint1 NA
## dat$Ltemp1:dat$LPres1:dat$Ftemp1:dat$Fdpoint1 NA
## dat$Ltime1:dat$LPres1:dat$Ftemp1:dat$Fdpoint1 NA
## dat$Ltemp1:dat$Ltime1:dat$Fctime1:dat$Fdpoint1 NA
## dat$Ltemp1:dat$LPres1:dat$Fctime1:dat$Fdpoint1 NA
## dat$Ltime1:dat$LPres1:dat$Fctime1:dat$Fdpoint1 NA
## dat$Ltemp1:dat$Ftemp1:dat$Fctime1:dat$Fdpoint1 NA
## dat$Ltime1:dat$Ftemp1:dat$Fctime1:dat$Fdpoint1 NA
## dat$LPres1:dat$Ftemp1:dat$Fctime1:dat$Fdpoint1 NA
## dat$Ltemp1:dat$Ltime1:dat$LPres1:dat$Ftemp1:dat$Fctime1 NA
## dat$Ltemp1:dat$Ltime1:dat$LPres1:dat$Ftemp1:dat$Fdpoint1 NA
## dat$Ltemp1:dat$Ltime1:dat$LPres1:dat$Fctime1:dat$Fdpoint1 NA
## dat$Ltemp1:dat$Ltime1:dat$Ftemp1:dat$Fctime1:dat$Fdpoint1 NA
## dat$Ltemp1:dat$LPres1:dat$Ftemp1:dat$Fctime1:dat$Fdpoint1 NA
## dat$Ltime1:dat$LPres1:dat$Ftemp1:dat$Fctime1:dat$Fdpoint1 NA
## dat$Ltemp1:dat$Ltime1:dat$LPres1:dat$Ftemp1:dat$Fctime1:dat$Fdpoint1 NA
##
## (Intercept) ***
## dat$Ltemp1 *
## dat$Ltime1 ***
## dat$LPres1 **
## dat$Ftemp1 ***
## dat$Fctime1 ***
## dat$Fdpoint1 ***
## dat$Ltemp1:dat$Ltime1 .
## dat$Ltemp1:dat$LPres1
## dat$Ltime1:dat$LPres1 **
## dat$Ltemp1:dat$Ftemp1
## dat$Ltime1:dat$Ftemp1 **
## dat$LPres1:dat$Ftemp1
## dat$Ltemp1:dat$Fctime1
## dat$Ltime1:dat$Fctime1
## dat$LPres1:dat$Fctime1
## dat$Ftemp1:dat$Fctime1
## dat$Ltemp1:dat$Fdpoint1
## dat$Ltime1:dat$Fdpoint1
## dat$LPres1:dat$Fdpoint1
## dat$Ftemp1:dat$Fdpoint1 ***
## dat$Fctime1:dat$Fdpoint1
## dat$Ltemp1:dat$Ltime1:dat$LPres1
## dat$Ltemp1:dat$Ltime1:dat$Ftemp1
## dat$Ltemp1:dat$LPres1:dat$Ftemp1
## dat$Ltime1:dat$LPres1:dat$Ftemp1 *
## dat$Ltemp1:dat$Ltime1:dat$Fctime1
## dat$Ltemp1:dat$LPres1:dat$Fctime1
## dat$Ltime1:dat$LPres1:dat$Fctime1
## dat$Ltemp1:dat$Ftemp1:dat$Fctime1
## dat$Ltime1:dat$Ftemp1:dat$Fctime1
## dat$LPres1:dat$Ftemp1:dat$Fctime1
## dat$Ltemp1:dat$Ltime1:dat$Fdpoint1
## dat$Ltemp1:dat$LPres1:dat$Fdpoint1
## dat$Ltime1:dat$LPres1:dat$Fdpoint1
## dat$Ltemp1:dat$Ftemp1:dat$Fdpoint1
## dat$Ltime1:dat$Ftemp1:dat$Fdpoint1
## dat$LPres1:dat$Ftemp1:dat$Fdpoint1
## dat$Ltemp1:dat$Fctime1:dat$Fdpoint1
## dat$Ltime1:dat$Fctime1:dat$Fdpoint1
## dat$LPres1:dat$Fctime1:dat$Fdpoint1
## dat$Ftemp1:dat$Fctime1:dat$Fdpoint1
## dat$Ltemp1:dat$Ltime1:dat$LPres1:dat$Ftemp1
## dat$Ltemp1:dat$Ltime1:dat$LPres1:dat$Fctime1
## dat$Ltemp1:dat$Ltime1:dat$Ftemp1:dat$Fctime1
## dat$Ltemp1:dat$LPres1:dat$Ftemp1:dat$Fctime1
## dat$Ltime1:dat$LPres1:dat$Ftemp1:dat$Fctime1
## dat$Ltemp1:dat$Ltime1:dat$LPres1:dat$Fdpoint1
## dat$Ltemp1:dat$Ltime1:dat$Ftemp1:dat$Fdpoint1
## dat$Ltemp1:dat$LPres1:dat$Ftemp1:dat$Fdpoint1
## dat$Ltime1:dat$LPres1:dat$Ftemp1:dat$Fdpoint1
## dat$Ltemp1:dat$Ltime1:dat$Fctime1:dat$Fdpoint1
## dat$Ltemp1:dat$LPres1:dat$Fctime1:dat$Fdpoint1
## dat$Ltime1:dat$LPres1:dat$Fctime1:dat$Fdpoint1
## dat$Ltemp1:dat$Ftemp1:dat$Fctime1:dat$Fdpoint1
## dat$Ltime1:dat$Ftemp1:dat$Fctime1:dat$Fdpoint1
## dat$LPres1:dat$Ftemp1:dat$Fctime1:dat$Fdpoint1
## dat$Ltemp1:dat$Ltime1:dat$LPres1:dat$Ftemp1:dat$Fctime1
## dat$Ltemp1:dat$Ltime1:dat$LPres1:dat$Ftemp1:dat$Fdpoint1
## dat$Ltemp1:dat$Ltime1:dat$LPres1:dat$Fctime1:dat$Fdpoint1
## dat$Ltemp1:dat$Ltime1:dat$Ftemp1:dat$Fctime1:dat$Fdpoint1
## dat$Ltemp1:dat$LPres1:dat$Ftemp1:dat$Fctime1:dat$Fdpoint1
## dat$Ltime1:dat$LPres1:dat$Ftemp1:dat$Fctime1:dat$Fdpoint1
## dat$Ltemp1:dat$Ltime1:dat$LPres1:dat$Ftemp1:dat$Fctime1:dat$Fdpoint1
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.002952 on 48 degrees of freedom
## Multiple R-squared: 0.849, Adjusted R-squared: 0.8018
## F-statistic: 18 on 15 and 48 DF, p-value: 9.012e-15Reg expression for reducing camber is as following,
Camber = 0.0157250 + 0.0010094(A) + 0.0017844(C) - 0.0041875(E) - 0.0077469(F)
Best strategy to reduce camber would be to keep A and C at low level and E and F at high level.
Q8.40)
a)
This experiment investigated “Four” Factors. They are A , B , C , D
b)
Resolution of the design employed in this experiment is “Four”.
c)
O = 8
AD =10
BD =12
AB =7
CD =13
AC = 6
BC =5
ABCD =11
EffectA <- (2*(AD+AB+AC+ABCD-O-BD-CD-BC))/(8)
EffectA## [1] -1EffectB <- (2*(BD+AB+BC+ABCD-O-AD-CD-AC))/(8)
EffectB## [1] -0.5EffectC <- (2*(CD+AC+BC+ABCD-O-AD-BD-AB))/(8)
EffectC## [1] -0.5EffectD <- (2*(AD+BD+CD+ABCD-O-AB-AC-BC))/(8)
EffectD## [1] 5Above we saw effect calculation from formula , lets confirm it from another way
a <- c(rep(c(-1,1),4))
b <- c(rep(c(-1,-1,1,1),2))
c <- c(rep(c(-1,-1,-1,-1,1,1,1,1),1))
d <- c(-1,1,1,-1,1,-1,-1,1)
resp <- c(8,10,12,7,13,6,5,11)
dataeff <- data.frame(a,b,c,d,resp)
modeleff <- lm(resp~a*b*c*d,data = dataeff)
coeff <- coef(modeleff)
effects <- coeff*2
effects## (Intercept) a b c d a:b
## 18.0 -1.0 -0.5 -0.5 5.0 1.5
## a:c b:c a:d b:d c:d a:b:c
## 0.5 -1.0 NA NA NA NA
## a:b:d a:c:d b:c:d a:b:c:d
## NA NA NA NA-1 is effect of A
-0.5 is effect of B
-0.5 is effect of C
5 is effect of D
Part D
Defining relation for this design is I = ABCD.
Q8.48)
a)
If we see the design generator for column D is -ABC.
b)
If we see the design generator for column E is BC.
c)
From table 8.14 from Douglas Montgomery Resolution of folded over design for such scenrio is “Four”.
8.60)
des <- FrF2(nfactors=7,resolution=3,randomize=FALSE)
des## A B C D E F G
## 1 -1 -1 -1 1 1 1 -1
## 2 1 -1 -1 -1 -1 1 1
## 3 -1 1 -1 -1 1 -1 1
## 4 1 1 -1 1 -1 -1 -1
## 5 -1 -1 1 1 -1 -1 1
## 6 1 -1 1 -1 1 -1 -1
## 7 -1 1 1 -1 -1 1 -1
## 8 1 1 1 1 1 1 1
## class=design, type= FrF2des2 <- fold.design(des,column=1)
des2## A B C fold D E F G
## 1 -1 -1 -1 original 1 1 1 -1
## 2 1 -1 -1 original -1 -1 1 1
## 3 -1 1 -1 original -1 1 -1 1
## 4 1 1 -1 original 1 -1 -1 -1
## 5 -1 -1 1 original 1 -1 -1 1
## 6 1 -1 1 original -1 1 -1 -1
## 7 -1 1 1 original -1 -1 1 -1
## 8 1 1 1 original 1 1 1 1
## 9 1 -1 -1 mirror 1 1 1 -1
## 10 -1 -1 -1 mirror -1 -1 1 1
## 11 1 1 -1 mirror -1 1 -1 1
## 12 -1 1 -1 mirror 1 -1 -1 -1
## 13 1 -1 1 mirror 1 -1 -1 1
## 14 -1 -1 1 mirror -1 1 -1 -1
## 15 1 1 1 mirror -1 -1 1 -1
## 16 -1 1 1 mirror 1 1 1 1
## class=design, type= FrF2.foldedtypeof(des2)## [1] "list"des3 <- des2[-c(1,3,5,7,10,12,14,16),]
des3## A B C fold D E F G
## 2 1 -1 -1 original -1 -1 1 1
## 4 1 1 -1 original 1 -1 -1 -1
## 6 1 -1 1 original -1 1 -1 -1
## 8 1 1 1 original 1 1 1 1
## 9 1 -1 -1 mirror 1 1 1 -1
## 11 1 1 -1 mirror -1 1 -1 1
## 13 1 -1 1 mirror 1 -1 -1 1
## 15 1 1 1 mirror -1 -1 1 -1aliasprint(des2)## $legend
## [1] A=A B=B C=C D=fold E=D F=E G=F H=G
##
## $main
## [1] B=CG=FH C=BG=EH E=CH=FG F=BH=EG G=BC=EF H=BF=CE
##
## $fi2
## [1] AB=-DE AC=-DF AD=-BE=-CF=-GH AE=-BD AF=-CD
## [6] AG=-DH AH=-DG#aliasprint(des3)